Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence{r(n)}, wherer(n) = E[x(l_{m+n}) overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if{t_n}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort leq t_oort geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known{ r(n) }.     

Energy Packing Efficiency of the Hadamard Transform

This concise paper presents a theoretical evaluation of energy packing of the Hadamard transform which is often used in signal processing. It is shown that energy contained in the lowest1/2^{j}(j: positive integer) of a signal's sequency spectrum can be explicitly evaluated in terms of covariances of the signal.     

On the mapping by a cross-correlation antenna system of partially coherent radio sources

In this paper it is shown that by means of a crosscorrelation antenna system it is possible to measure the mutual power distribution,T(u, v), of partially coherent radio sources. The two linear antennas of the system are scanned independently to give a two-dimensional output which can be written asbar{R}(u, v) = T(u, v) ast A(-u)B^{ast}(-v)whereA(u)andB(v)are the patterns of the two antennas. A two-dimensional Fourier analysis of the output shows it to be a smoothed version of the true mutual power distribution. One can define a principal solution,T_{0}(u, v), which is a generalization to two dimensions of the well-known principal solution,T_{0}(u), which occurs when the sources are incoherent. The limiting cases of complete coherence and complete incoherence are considered. It is shown that for coherent sources the principal solution is factorable,T_{0}(u, v)=varepsilon_{0}(u)varepsilon_{0}^{ast}(v). For incoherent sources it is shown that the spatial frequency spectrum ofT_{0}(u, v)is a function of only the difference,x-y, of the spatial co-ordinates,t_{0}(x, y) = t_{0}(x- y).     

On the detection of known binary signals in Gaussian noise of exponential covariance

The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted byq(t), of a given integral equation is well behaved(L_{2}). For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with aq(t)which is notL_{2}is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noiseR(tau) = sigma^{2} e^{-alpha|tau|}is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic whenq(t)is notL_{2}, it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employsalpha_{infty}(which in general depends on the observed sample function) in place ofalpha, the operations of the optimum detector are unchanged by the uncertainty inalpha. It is then shown that almost all sample functions can be used to yield a perfect estimate ofalpha. Using this estimate ofalpha, a test statistic equivalent to the Neyman-Pearson statistic is obtained.     

On estimating spectral moments in the presence of colored noise

Let{q^(1) (t)}, the signal, be a complex Gaussian process corrupted by additive Gaussian noise{q^(2) (t) }. Observations onp(t)q(t)andp(t) q^(2) (t)are assumed to be available wherep(t)is a smooth weighting function andq = q^(1) + q^(2). Using the Fourier transform of the samples ofp(t)q(t)andp(t) q^(2) (t), estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.     

On the capacity of certain additive non-Gaussian channels

A model of an additive non-Gaussian noise channel with generalized average input energy constraint is considered. The asymptotic channel capacityC_{zeta}(S), for large signal-to-noise ratioS, is found under certain conditions on the entropyH_{ tilde{ zeta}}( zeta)of the measure induced in function space by the noise processzeta, relative to the measure induced bytilde{zeta}, where is a Gaussian process with the same covariance as that ofzeta. IfH_{ tilde{zeta}}( zeta) < inftyand the channel input signal is of dimensionM< infty, thenC_{ zeta}(S)= frac{1}{2}M ln(1 + S/M) + Q_{zeta}( M ) + {o}(1), where0 leq Q_{ zeta}( M ) leq H_{ tilde{ zeta}}( zeta). If the channel input signal is of infinite dimension andH_{ tilde{ zeta}}( zeta) rightarrow 0forS rightarrow infty, thenC_{ zeta}(S) = frac{1}{2}S+{o}(1).     

The propagation of stochastic waves in uniform media

A solution is presented for the correlation function (mutual coherence) and time average power of a stochastic wave in the uniform half-spacez geq 0, given as a boundary value the correlation function on the planez = 0. Stationary statistics are assumed and solution is effected via a spectral representation of the field. It is found that when loss is present the evanescent spectrum, as defined for the lossless case, contributes to energy propagation; and criterion are developed which determine when the integral of spectral density can be associated with power.     

A new approach to the problem of wave fluctuations in localized smoothly varying turbulence

In the past, smoothly varying turbulence has been studied by changing the structure constant to the functionC_{n}^{2}(bar{r}). The purpose of this paper is to show that this approach is insufficient, and that a random process developed by Silverman can be used to describe the wave fluctuations in localized smoothly varying turbulence. The localized turbulence is characterized by a correlation function which is a product of a function of the average coordinate and a function of the difference coordinate. The corresponding spectrum is also given by a product of a function of the difference wavenumber and a function of the average wavenumber. They are related to each other through two Fourier transform pairs. Making use of the preceding representations, the fluctuations of a wave propagating through such a turbulence can be given either by the integrals with respect to the two wavenumbers or by a convolution integral of the structure constantC_{n}^{2}(bar{r}) and a function involving the outer scale of the turbulenceL_{0}. It is shown that for a plane wave case, if the distanceLis within (L_{0}^{2}/lambda), then the usual formula given by Tatarski is valid. But if the distance is betweenL_{0}^{2}/lambdaand(bL_{0})/lambdawherebis the total transverse size of the turbulence, the variance of the wave is nearly constant, and ifL gg (bL_{0})/lambda, the variance decays asL^{-2}. Similar conclusions are shown for a spherical wave case. Some examples are shown illustrating the effectiveness of this method.     

The correlation function of Gaussian noise passed through nonlinear devices

This paper is concerned with the output autocorrelation functionR^{y}of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes inR^{y}when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governingR^{y}is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devicesy = x^{m}withmeither an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculatingR^{y}.     

Magnitude and dispersion of Kleinman forbidden nonlinear optical coefficients

Starting with a perturbation expansion for the Kleinman forbidden nonlinear optical coefficientd_{ijk^{F}}and for Miller'sDelta_{ijk^{F}}, and making several approximations, we arrive at a simple result for the ratio of forbidden to allowed mixing nonlinearities (omega_{1} + omega_{2} = omega_{3}), namelyDelta_{ijk^{F}}/Delta_{ijk^{A}} propto (omega_{3}^{2} + 2omega_{1}omega_{2}). For second-harmonic generation (SHG) this can be expressed asDelta_{ijk^{F}}/Delta_{ijk^{A}} simeq (omega/chi)(partialchi/partialomega), which clearly shows the close connection betweenDelta_{ijk^{F}}and the linear dispersion. These expressions are shown to give good agreement with literature experimental values, as well as for our measurements on TeO2for various input frequencies ω1and ω2(i.e.,omega_{3} = 1.88, 2.33, 2.82, 3.50, and 3.76 eV).     

Nonlinear coupled transmission line analysis for second harmonic generation in traveling-wave GaAs MESFET's

A coupled nonlinear transmission line analysis for second harmonic generation from a traveling-wave field-effect transistor is presented. In the formulation, three main nonlinearities, namely the Schottky-barrier capacitance, the transconductance and the draincon-ductance, are considered. The coupled nonlinear differential equations are solved under a weak-coupling approximation with the help of the Weierstrassian elliptical function. It is observed that the second harmonic gain is significant afterx/l ≃ 0.7(wherelis the transverse length of the transistor). After this value of x, the amplitude at second-harmonic level increases sharply and follows almost a square-law behavior. It is found that the output drain conductance emerges as a large contributor to harmonic generation. The possibility of an image gate as drain conductivity modulator or phase velocity synchronizer is proposed. It is confirmed that travelling-wave MESFET's can also be used as harmonic generators.     

Random ciphering bounds on a class of secrecy systems and discrete message sources

The problem of enciphering a stationary finite discrete message so that a cryptanalyst is unlikely to decrypt an intercepted cryptogram is considered. Additive-like instantaneous block (ALIB) encipherers are studied that employ a list ofe^{nr}keywords of lengthn, called the cipher. An ALIB encipherer produces a cryptogram word of lengthnfrom a message word and a key word of the same length by combining corresponding message letters and key-word letters. Certain technical restrictions sure placed on the combining function. The decipherer uses a decoder which combines a letter from the key word used in enciphering with a letter from the cryptogram to form a letter of the decoded message. cryptanalyst also decodes letter by letter with an identical decoder; however, he uses a keyword that is not necessarily that used in enciphering. For a given message source and combiner, the design of the cipher consists in choosing the block lengthn, the key rater, and the set ofe^{nr}key words. These are to be chosen so thatp_{w}, the probability of correct decryptment of the message word, andp( Delta), the probability that the per letter nonzero Hamming distance between the decrypted message and the true message is smaller thanDelta, are very small for every cryptanalyst. A set of pairs( Delta,r)for which there exist ciphers with key ratersuch that,p_{w}andp( Delta)can be made arbitrarily small for every cryptanalyst is determined using the concepts of random ciphering and exponential bounding.     

Complex-correlation radiometer

The correlation between the signals from the two antennas in a radio interferometer can be described by a complex correlation function. The real and imaginary parts of this function can be measured by a radiometer using two feedback loops to control four noise sources,B_{pr}(positive, real),B_{nr}(negative, real),B_{pi}(positive, imaginary), andB_{ni}(negative, imaginary). Theoretical analysis shows the system to be stable against amplifier gain fluctuations, and against moderate fluctuations in the phase shifts of the two amplification channels which are included in the system. Experimental verification is described in a following paper by Hubbard and Erickson [2].     

Analysis of a cross correlator with a clipper in one channel (Corresp.)

General expressions are derived for the characteristic function and probability density function of the output of a cross crrelator utilizing a hard clipper in one input channel. The inputs are assumed to beA_{1} cos(omega_{o}t)+n_{l}(t)andA_{2} cos(omega_{o}t+phi)+n_{2}(t), wheren_{1}andn_{2}are independent narrowband Gaussian noises. From the characteristic function, which is expressed in the form of a Fourier series in the relative phase anglephi, a double series involving cosines of multiples ofphiand Hermite polynomials of all orders is obtained as the probability density function of the output. Both the characteristic function and the density function pertain to the output at only a single instant of time. The first and second moments of the output are also calculated from the characteristic function, and lead to a closed-form expression for the output signal-to-noise ratio as a function of the input signal-to-noise ratios. Limiting cases of the output signal-to-noise ratio are computed and compared with similar cases involving the more conventional analog correlator without clippers and with the polarity-coincidence correlator.     

Bent-function sequences

In this paper we construct a new family of nonlinear binary signal sets which achieve Welch's lower bound on simultaneous cross correlation and autocorrelation magnitudes. Given a parameternwithn=0 pmod{4}, the period of the sequences is2^{n}-1, the number of sequences in the set is2^{n/2}, and the cross/auto correlation function has three values with magnitudesleq 2^{n/2}+1. The equivalent linear span of the codes is bound above bysum_{i=1}^{n/4}left(stackrel{n}{i} right). These new signal sets have the same size and correlation properties as the small set of Kasami codes, but they have important advantages for use in spread spectrum multiple access communications systems. First, the sequences are "balances," which represents only a slight advantage. Second, the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly "hopped" from sequence to sequence for code division multiple access operation. Most importantly, the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it. This high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing.     

Toward a theory of unknown functions (Corresp.)

A functionf(x)is chosen from a finite set of functionscal F. An outsider observer knowscal Fbut not the actual choicef(x). He is, however, able to make a limited number of observations(x,y)satisfying the unknown functiony = f(x). The uncertainty of the outside observer with respect to the unknown function is measured as the entropy of the output variableywhen the functionf(x)is regarded as a random choice incal F. With this measure, an upper bound on the uncertainty is derived. The existence of unknown functions satisfying this bound is investigated, and necessary and sufficient conditions are derived. The problem is shown to be closely related to the problem of finding algebraic codes with high minimum Hamming distance. The theory can be applied to cryptography, identifying mechanisms, access control in computers, and possibly also to reliability analysis.     

Properties ofPN^2sequences (Corresp.)

A method for generating sequences that approximate binary random sequences with the probability of a 1 equal to 1/4 is described. These are calledPN^2sequences.PN^2sequences are generated by clocking aPNsequence generator at the l's of aPNsequence. ThePN^2sequence is 1 when the generator output makes a transition and is 0 otherwise. It is shown thatPN^2sequences have periodN^2if thePNsequence generators have periodN. The density of l's is shown to approach 1/4 for largeN. It is shown that the normalized out-of-phase pulse-coincidence autocorrelation function can never exceed 1/2 or be less than 1/4 and is 1/4 most of the time for largeN.     

Lack of Spatial Correlation in MOSFET Threshold Voltage Variation and Implications for Voltage Scaling

Due to increased variation in modern process technology nodes, the spatial correlation of variation is a key issue for both modeling and design. We have created a large array test-structure to analyze the magnitude of spatial correlation of threshold voltage $(V_{T})$ in a 180 nm CMOS process. The data from over 50 k measured devices per die indicates that there is no significant within-die spatial correlation in $V_{T}$. Furthermore, the across-chip variation patterns between different die also do not correlate. This indicates that Random Dopant Fluctuation (RDF) is the primary mechanism responsible for $V_{T}$ variation and that relatively simple Monte Carlo-type analysis can capture the effects of such variation. While high performance digital logic circuits, at high $V_{DD}$ , can be strongly affected by spatially correlated channel length variation, we note that subthreshold logic will be primarily affected by random uncorrelated $V_{T}$ variation.      

On the distribution and moments ofRC-filtered hard-limitedRC-filtered white noise

Suppose white noise is put into anRCfilter with time constant1/awhose output is hard limited and fed into a secondRCfilter with time constant1/b. The density of the outputy(t)of the system has been the subject of several investigations. Most recently, Pawula and Tsai made the conjecture that for Gaussian white noise they(t)density is given by a certain expression, a special case of which was derived earlier by Doyle, McFadden, and Marx forb/a = 2. In this paper an expression for thenth moment ofy(t)is found. From the expression for the fourth moment, it is proven that the Pawula-Tsai conjecture is not true in general. It is further found that if the output of the hard limiter were wide-sense Markov, the Pawula-Tsai conjecture would be true.     

Harmonic analysis for a class of multiplicative processes

The harmonic analysis of certain multiplicative processes of the formg(t)X(t)is considered, wheregis a deterministic function, and the stochastic processX(t)is of the formX(t)=sum X_{n}l_{[n alpha , (n+l) alpha]}(t), where a is a positive constant and theX_{n}, n=0, pm 1,pm 2, cdotsare independent and identically distributed random variables with zero means and finite variances. In particular, we show that if g is Riemann integrable and periodic, with period incommensurate withalpha, theng(t)X(t)has an autocovariance in the Wiener sense equal to the product of the Wiener autocovariances of its factors,C_{gx} = C_{g}C_{x}. Some important cases are examined where the autocovariance of the multiplicative process exists but cannot be obtained multiplicatively.